Equation (REF ) must be accompanied by a relation
between saturation \(s\)  and pressure \(p\) . Models without hysteresis
demand either the algebraic relation \(p = p_c(s)\)  for some given
function \(p_c:[0,1] \rightarrow \bar{\mathbb {R}}\) , or they include the
“\(\tau \) -correction” and demand, for some physical parameter
\(\tau >0\) , known as the dynamic capillary number, that
\(p=p_c(s)+ \tau \partial _t s\) ; this latter model takes inertia in the
material law into account, see [1]}.
If, additionally, hysteresis in an imbibition process shall be
modelled, a possible simple law is
\(\partial _t s=\frac{1}{\tau }[p-p_c(s)]_+\,,\) 
